The Smoothing Parameter

The choice of the smoothing parameter $\lambda$ is critical to the performance of a spline estimate. Several data-adaptive methods have been successfully used in practice (Wahba, 1990). The following three methods, Generalized Cross Validation (GCV), Generalized Maximum Likelihood (GML) and Unbiased Risk (UBR), were implemented in RKPACK, and are available in S functions dsidr and ssr. Denote $A(\lambda)$ as the hat matrix such that

$$(L_1\hat{f}_{\lambda},\cdots,L_n\hat{f}_{\lambda})^T=A(\lambda) \mbox{\boldmath $y$}.$$

The GCV, GML and UBR methods estimate $\lambda$ as the minimizers of the following GCV function

$$\mbox{GCV}(\lambda)=\frac{\displaystyle \frac{1}{n} \vert\v... ...t^2}{\displaystyle [\frac{1}{n}\mbox{tr}(I-A(\lambda))]^2},$$

GML function

$$\mbox{GML}(\lambda)=\frac{\mbox{\boldmath $y$}^T(I-A(\lambda... ...{\boldmath $y$}} {[\mbox{det}^+ ((I-A(\lambda)))]^{1/(n-M)}},$$

where $\mbox{det}^+$ represents the product of the nonzero eigenvalues, and UBR function

$$U(\lambda)=\frac{1}{n}\vert\vert(I-A(\lambda))\mbox{\boldmath $y$}\vert\vert^2+ \frac{2\sigma^2}{n}\mbox{tr}A(\lambda),$$

respectively.

The GCV method may lead to interpolation when the sample size is small (Wahba and Wang, 1993). The GML method is very stable. For moderate sample sizes, the performance of the GCV and GML methods are comparable. For large sample sizes, the GCV method performs better then the GML method. In our S function ssr, an option spar is provided for specifying one of these three methods. spar=``v'', spar=``m'' and spar=``u'' correspond to the GCV, GML and UBR methods respectively with GCV as the default. For example, fitting a cubic spline with the GML choice of the smoothing parameter can be accomplished by

    ssr(y~t, rk=cubic(t), spar=``m'')

An estimate of $\sigma^2$ is needed for the UBR method. It can be specified with the argument varht. For example, the following statement uses UBR to choose the smoothing parameter with $\sigma^2=10$

    ssr(y~t, rk=cubic(t), spar=``u'', varht=10)

Several methods may be used to derive an estimate of $\sigma^2$ (Donoho and Johnston, 1994; Gasser et al., 1986; Hall et al., 1990; Rice, 1984; Dette et al., 1998).